On the same system, using GetKeyState on a keyboard key continues to work while the window is inactive
Just doing a loop with Tooltip % A_TickCount continues to work while the window is inactive
So this seems to be solely related to the XBE / XBA controllers

DirectInput (like WinMM) has a limitation regarding the triggers on XInput controllers, while XInput only supports a much more limited subset of buttons and axes (and controllers). DirectInput is useless to me because I only use XInput controllers, and built-in XInput support would not be a sound investment of my time because I haven't used a controller with AutoHotkey in three years, and XInput.ahk already does everything I needed.

These are the alternatives that I know about for XInput (Xbox 360 and Xbox One) controllers:

@OneSwitch3000 - This was not really about trying to get XInput support built in to AHK, I was just trying to ascertain what the root cause was and whether there was a simple fix.
I think it would be much less work for me to implement XInput in UCR than for Lexikos to integrate with AHK, and I am not sure there is a huge amount of advantage in the latter.
I had hoped to not put any more dev effort into UCR-AHK, instead focusing solely on UCR-C#, but migrating you away from UCR-AHK is not going to happen in the very short term, so I am going to have to bite the bullet I suppose - I am looking into integrating XInput into UCR-AHK now.